(3) Magnetization Process

(3) Magnetization Process

 Uniaxial Magnetization

Hard-axis Magnetization Easy-axis Magnetization

 Field at Arbitrary Orientation to Uniaxial Easy Axis

Stoner-Wohlforth Problem

Magnetization Change by Curling Free Domain Walls

Approach to Saturation

 Domain Wall Pinning and Coercivity

Large “Fuzzy” Defects

Micromagnetic Theory for Well-defined Defects Examples

Additional Anisotropy

 Uniaxial Magnetization

(Origin of uniaxial anisotropy : magnetostatic, magnetocrystalline, magnetoelastic, or field-induced)

K

= (1/2)μ

(N

– N

)M

for magnetostatic (shape) K

for magnetocrystalline

3λ

σ/2 for magnetoelastic (isotropic)

Hard-axis

magnetization :

Related magnetic energy - Uniaxial anisotropy energy - Zeeman energy

Easy-axis

magnetization :

Related magnetic energy - Uniaxial anisotropy energy - Zeeman energy

(3) Magnetization Process

Hard-axis Magnetization

f = f_a + f_Zeeman = K_usin²(π/2 -θ) – M_s•B

= K_ucos²θ – M_sHcosθ, K_u > 0 Torque, T_θ = – ∂f/∂θ on M

Minimum f when ∂f/∂θ = (– 2K_ucosθ + M_sH)sinθ = 0 and ∂²f/∂θ² = – 2K_ucos²θ + M_sHcosθ > 0

Zero torque: sinθ = 0 → θ = 0, π, …

Stability conditions (see Fig. 2.12 in O’Handley) : θ = 0 is stable only for H > 2K_u/M_s (K_u > 0)

θ = π is stable only for H < – 2K_u/M_s (K_u > 0)

Equation of motion for the magnetization in fields below saturation ( – 2K_u/M_s < H < 2K_u/M_s) 2K_ucosθ = M_sH

For H = 0, θ = π/2, H_a (anisotropy field) = H_satfor cosθ = 1

Since M = M_scosθ, K_u = M_sH_a/2, Then 2K_ucosθ = M_sH → H_aM_scosθ = M_sH Using reduced magnetization, m = M/M_s = cosθ and h = H/H_a

m = h for – 1 ≤ h ≤ 1 (see Fig. 9.2(a) in O'Handley)

(3) Magnetization Process

Easy-axis Magnetization (see Fig. 9.1(b) in O'Handley) Free energy in a single domain particle (or completely pinned wall)

f = - M_sHcosθ + K_usin²θ

Energy density f versus θ for various h(reduced field) = M_sH/2K_u Stability conditions:

For h = 0 (i.e., zero field), stable at both θ = 0 and π

For 0 < h < 1, angular position of min. f is independent of field.

For h = 1, θ = π is no more stable, m = 0 called the switching field.

Zero-torque condition: M_sHsinθ + K_usin2θ = 0 θ = 0 (m = 1) and π (m = – 1)

Stability conditions: M_sHcosθ + 2K_ucos2θ > 0 θ = 0 is stable only for H > -2K_u/M_s(h > – 1) θ = π is stable only for H < 2K_u/M_s(h < 1)

θ = 0 and π are only locally stable for -2K_u/M_s < H < 2K_u/M_s In Fig. 9.2(b), solid lines : stable solutions

dashed lines : locally stable solutions → square hysteresis loop!!!

(3) Magnetization Process

Easy-axis Magnetization (continued)

In the case of domain wall motion :

No torque on the domain magnetization, but a torque on the spins making up the wall - The spins in the wall may rotate to align with H

Macroscopically, the difference in Zeeman energy of the two domains (a field-induced potential energy difference) across the domain wall (= 2M_sH) lower energy by moving so as to reduce the volume of the unfavorably oriented domain

Equivalently, the force on the domain wall, given by F = -

∂U/∂x

-Assuming smooth and easy wall motion (negligible pinning), solid line with zero coercivity (see Fig. 9.2(b) in O’Handley)

Coercive field H_c :

H_c = 2K_u/M_s in the single-domain or pinned wall limit by rotational hysteresis H_c = 0 in free-domain-wall limit

Permeability μ_rot for a purely rotational magnetization process :

2K_ucosθ = M_sH → H_aM_scosθ = M_sH since H_c = 2K_u/M_s and M = M_scosθ → M = (M_s²/2K_u)H μ_rot= μ_o(H +M)/H = μ_o(1 + M_s²/2K_u) ≈ μ_oM_s²/2K_u(SI unit) μ_rot ≈ 2πM_s²/K_u (cgs)

If K_uis small, the M-H loop is steep and μ_rot can be large!!!

(3) Magnetization Process

Stoner-Wohlfarth Problem :

The magnetization process for single-domain particles of ellipsoidal shape

- The M-H loops for a field applied at an arbitrary angle θ_o with respect to a uniaxial easy axis

(see Fig. 9.4 in O'Handley)

For a prolate spheroid, the free energy f f = – K_ucos²(θ – θ_o) – M_sHcosθ

where K_u= [H_a+ (N₂- N₁)M_s]M_s/2, H_a is the anisotropy field, N₁ and N₂ are demagnetization factors // and 

to the easy axis of the particle, respectively.

Minimum free energy when ∂f/∂θ = 0

2K_usin(θ - θ_o)cos(θ - θ_o) + M_sHsinθ = 0

Using K_u = H_aM_s/2 and the reduced field h = H/H_a, sin2(θ - θ_o) + 2hsinθ = 0

With the reduced magnetization m = M/M_s= cosθ, the solution can be written as the following,

2m(1 - m²)^1/2cos2θ + (1 - 2m²)sin2θ ± 2h(1 - m²)^1/2= 0 → h = f(m) (see Fig. 9.5 in O'Handley)

 Field at Arbitrary Orientation to Uniaxial Easy Axis

(3) Magnetization Process

Equation of motion for the magnetization :

2m(1 - m²)^1/2cos2θ_o + (1 - 2m²)sin2θ_o ± 2h(1 - m²)^1/2 = 0 Interpretations

(i) θ_o= π/2 : Hard-axis magnetization (Fig. 9.2(a) in O'Handley) (ii) θ_o= 0 : Easy-axis magnetization

(Fig. 9.2(b) in O'Handley)

(iii) Single-domain, oblique magnetization process in negative fields

- For a small range of θ_oabove 0^o,

the magnetization reversal occurs at a critical field, h_s = H_s/H_a, called the reduced switching field;

h_s is defined where m(h) curve satisfies ∂h/∂m = 0.

At the switching point,

the magnetization switches abruptly and reversibly.

free energy minimum (∂f/∂θ = 0) becomes flat (i.e., unstable): ∂²f/∂θ² = 0.

h_scosθ - cos2(θ - θ_o) = 0 (since ∂f/∂θ = sin2(θ - θ_o) + 2hsinθ )

(3) Magnetization Process

Explanations (continued)

At the switching point,

h_scosθ - cos2(θ - θ_o) = 0 Then

sin2θ_o = (2/h_s²)[( 1 - h_s²)/3]^-3/2 Solving for the switching field,

h_s= (cos^2/3θ_o + sin^2/3θ_o)^-3/2 (see Fig. 9.6 in O'Handley)

For 45^o < θ_o < 90^o, h_s occurs after the magnetization has changed sign; the coercivity h_c is less than h_s. And thus h_c is defined at m = 0, h_c= sinθ_ocosθ_o

(3) Magnetization Process

Magnetization Change by Curling : -

(i) Coherent rotation of all moments in unison (see Fig. 9.7(a) in O'Handley)

(ii) Curling (see Fig. 9.7(b) in O'Handley) (iii) Buckling (see Fig. 9.7(c) in O'Handley)

(iv) Fanning (in chain of spheres) (see Fig. 9.7(d) in O'Handley) (v) Domino effect (see Fig. 9.7(e) in O'Handley)

-

By having fewer spins pointing away from the easy axis at any

given stage of the reversal process, exchange energyis raised and the magnetostatic energy is lowered.

Switching field H_s for magnetization curling in an elongated, single-domain particle of semiminor axis b is reduced from the uniform rotation value, H_c = (N_b- N_a)M_s, by replacing the hard-axis magnetostatic energy with the curling energy,

H_s= [(a/2)(b_o/b)²- N_a]M_s∝C₁/b²- C₂

where b_o= r_o/N_b^1/2is the single-domain radius for an ellipsoid of revolution and a is a function of the aspect ratio of particles (1.08 (infinite cylinder) ≤ a ≤ 1.42 (sphere)).

θ_odependence of H_s due to curling for various values of S = b/b_o, assuming no magnetocrystalline anisotroopy (see Fig. 9.8 in O'Handley).

(3) Magnetization Process

- For smaller particles of superparamgnetism, H_c∝C₁- C₂/b^3/2

As the particle volume decreases in the

superparamagnetic regime, H_c drops at a given temperature. (see Fig. 9.9 in O'Handley)

(3) Magnetization Process

Free Domain Walls :

For large particles (or polycrystalline sample packed densely) with domain walls

- Assuming completely free domain walls : H_c = 0 M-H characteristic for H > 0 is the uniform

rotation process

(see Fig. 9.10 in O'Handley) Explanations :

- With decreasing H form positive saturation, domain walls are nucleated and move easily once H < 0.

- Both magnetization rotation and domain wall motion may contribute to the M-H process.

- No magnetization reversal by abrupt rotational switching because of the easy wall motion.

(3) Magnetization Process

Approach to Saturation :

- Experimental determination of M_s is not always evident. In that case, the mathematical form of the approach to saturation is helpful. At T well below T_c

M(H) = M_s(1 - a/H) + χ_hfH

Here χ_hfis the high-field susceptibility and the term - aM_s/H accounts for rotation of M away from the applied field as H decreases. (see Fig. 9.11 in O'Handley)

- For single-domain particles, from the Stoner- Wohlfarth model, at the limit h>> 1 , m ≈ 1 (or 2m² – 1 ≈ 1)

± 2h(1 – m²)^1/2= (2m² – 1)sin2θ_o leading to m ≈ (1 – sin²2θ_o/4h²)^1/2

and thus M(H) = M_s(1 – H_a²sin²2θ_o/8H²) By extrapolating of M(H) vs H^-2, M_s can be

(3) Magnetization Process

Why are domain wall motions

irreversible in real materials under an external field H

?

Domain wall energy can be lowered at a particular position in the

material due to the defects such as grain boundaries, precipitates, inclusions, surface roughness, and other defects, and thus the wall motion can be effectively pinned or inhibited.

 Domain Wall Pinning and Coercivity

(3) Magnetization Process

- Two classes of defect (see Fig. 9.12 in O'Handley)

(i) A nonmagnetic inclusion or planar defect coincident with a wall : The need for a moment rotation (or a twist in M) across the defects is eliminated and thus the total wall energy can be locally reduced by σ_dw times the common cross-sectional area of the defect and wall.

(ii) A magnetic defect having a strong anisotropy(crystalline or magnetoelastic) relative to that of the matrix : Local wall energy is increased and thus a barrier to a domain wall motion can be posed effectively.

- Distribution of defects in a material

The domain wall potential σ_dw(x) is highly irregular with position.

The presence of the pinning defects leads to an irregular domain wall motion consisting of a series of Barkhausen jumps.

- Quantitative models of domain wall motion

Two regimes based on the ratio of defect size D to wall thickness δ_dw = π(A/K_u)^1/2. For D >> δ_dw, domain wall pinning on defects

(i) Large fuzzy defect case (ii) Sharply defined defect case

(3) Magnetization Process

Large fuzzy defects

O'Handley) : strain fields or composition fluctuations - The domain wall can be considered to be moving in an irregular but slowly changing potential

- The driving pressure due to the difference in Zeeman energy across a 180^owall : 2M_sH

- Resistance to the wall motion due to the gradient in the wall energy density : P = - dσ_dw/dx

Then, since σ_dw= 4(AK_u)^1/2 and K_u = K_xtl + (3/2)λ_sσ dσ_dw/dx = 4d(AK_u)^1/2/dx = 2[(π/δ_dw)∂A/∂x +

(δ_dw/π)(∂K_xtl/∂x + (3/2)λ_s∂σ/∂x]

Explanations :

- Spatial variations in exchange and anisotropy energies, determining the wall energy, exert the impeding force to the domain wall motion.

(3) Magnetization Process

Large fuzzy defects (continnued)

The steepest gradient in wall energy density can be taken to be the magnetic pressure responsible for the coercivity H_c

(dσ_dw/dx)_max = 2M_sH_c → H_c= (dσ_dw/dx)_max/2M_s

If the variation ~ D, and assuming a linear gradient, dσ_dw/dx = △σ_dw/D H_c≈ (2H_a/)(δ_dw/D)[△A/A + △K_xtl/K_xtl + (3/2)λ_s△σ/K_xtl]

H_c∝δ_dw/D times a sum of fluctuation terms expressing local variations in exchange stiffness, crystal anisotropy and magnetoelastic anisotropy, respectively.

where the anisotropy field,

H_a = 2K_xtl/M_s : upper limit to the coercivity

Coercive mechanism arising from the magnetostatic energy of defects proportional to △M/M H_c∝ △(2πM_s²)/2πM_s² = 2△M_s/M_s

(3) Magnetization Process

Micromagnetic theory for well-defined defects

Sharply defined defect case : grain boundaries, voids, antiphase boundaries, and dislocations

Total energy of the system in this model = exchange energy + uniaxial anisotroopy energy + Zeeman energy densities

A_i(∂θ/∂x)² + K_isin²θ - M_sHcosθ

where θ = angle between magnetization and easy axis (y axis) Over all space, minimum energy when

-2A_i(∂θ/∂x)² + K_isin²θ - M_sHcosθ = C_iθ

where i = 1, 2, 3 corresponding to the regions in Fig. 9.14 Boundary conditions

dθ/dx = 0 at x = ± ∞ and

θ = 0 at x = + ∞ and θ = π at x = - ∞

Therefore, C₁= - HM_1, C₃= + HM₁, C₂from continuity of θ and exchange torque A_idθ/dx at the interfaces x₁ and x₂

Reduced coercive field h_c = H_cM₁/K₁ (see Fig 9.15 and 9.16 in O'Handley) Summary (see Fig. 9.17 in O'Handley)

(3) Magnetization Process

(3) Magnetization Process

Examples

- NdFeB (see Fig 9.18 in O'Handley)

- Amorphous and crystallized Co-Nb-B alloys (see Fig 9.19 in O'Handley)

- Sm₂(Co, Cu, Fe, Zr)₁₇ (see Fig 9.17 in O'Handley)

-Nanoparticles (Chap 12 in O'Handley)

Additional Anisotropy

In some large single-domain particle systems (e.g., Fe in a SiO₂ matrix), the coercivity can exceed the single-domain limit (2K/M_s) due to an additional anisotropy (probably, stress or interfacial spin pinning) induced by the particle interaction with its nonmagnetic matrix.

(3) Magnetization Process

The area inside a B-H loop = the energy loss per unit volume in one cycle

Energy W dissipated in a toroidal core over one cycle = the integral of the power loss over a period:

W =

Ampere's law and Faraday's law, V(t) = - dφ/dt = - AdB/dt

W = : DC hysteresis loss

Classical eddy-current loss

Eddy-current loss about a single domain wall Multiple domain walls



t

i t V t dt

0

( ) ( )





dt ^ lA H t dB dT

H dB lA

t

( )

0

 AC Processes

 Microwave Magnetization Dynamics & Ferromagnetic Resonance

How does the magnetization of a ferromagnetic material respond when the drive-field frequency approaches the natural precession frequency of the moment in a magnetic field?

(3) Magnetization Process